Editing Equality (mathematics) (section)

=== Set equality based on first-order logic with equality ===
In first-order logic with equality (See {{Section link||Axioms}}), the axiom of extensionality states that two sets that ''contain'' the same elements are the same set.<ref>{{harvnb|Kleene|1967|page=189}}. {{harvnb|Lévy|2002|page=13}}. {{harvnb|Shoenfield|2001|page=239}}.</ref>
* Logic axiom: <math>x = y \implies \forall z, (z \in x \iff z \in y)</math>
* Logic axiom: <math>x = y \implies \forall z, (x \in z \iff y \in z)</math>
* Set theory axiom: <math>(\forall z, (z \in x \iff z \in y)) \implies x = y</math>

The first two are given by the substitution property of equality from first-order logic; the last is a new axiom of the theory. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by [[Azriel Lévy]].
: "The reason why we take up first-order predicate calculus ''with equality'' is a matter of convenience; by this, we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic."<ref>{{harvnb|Lévy|2002|page=4}}.</ref>